Primes in Spirals
By
Cassie Gribble
A Thesis Submitted to the Faculty of the
DEPARTMENT OF MATHEMATICS
In Partial Fulfillment of the Requirements
For the Degree of
MASTER OF MIDDLE SCHOOL MATHEMATICS LEADERSHIP
In the Graduate College
THE UNIVERSITY OF ARIZONA
2010
Abstract
Prime numbers have been a fascination to mathematicians for ages. They seem to come
in particular form. They do not seem to be distributed in any particular way. There is no
one formula that will generate all prime numbers. They do not seem to follow any
particular pattern, except maybe when looking at the prime numbers arranged in a
spiral.
2
Identifying Primes
Before looking at primes in spirals, you have to understand prime numbers, what
they are, and how to identify them. Prime numbers are positive whole numbers that have
only two factors, itself and one. An easy way to identify prime numbers is using the
Sieve of Eratosthenes. If using a hundreds chart, begin with two and cross out all of the
multiples of two, then continue with three and cross out all multiples of three not already
crossed out, and the next consecutive numbers not crossed out will be five then seven.
Once you have completed the multiples through seven, you can stop, only primes will be
left. Always crossing out the multiples, but not the beginning number. When you have
completed this process you are left with only prime numbers, which are highlighted
below.
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Figure 1. Primes using Sieve of Eratosthenes
Another way to identify prime numbers is using modulus six. To find primes
consider each multiple, m, of six and one greater and one less (mod 6+1 and mod 6+5) as
possible primes. Test each possibility by dividing it by numbers that are ≤√m. All of
the red numbers below are primes, with each arm being modulo 6 plus the number inside
the circle.
3
Figure 2. A modulus-6 clock spiral showing the primes (red) to 90
http://www.chass.utoronto.ca/french/as-sa/ASSA-14/article7en.html
Rectangular Array
My task was to see what the longest string of prime numbers I could find in a
regular rectangular array. I could change the width of the array or what I was counting
by. All of the equations for these arrays are linear. There is a constant difference
between each prime in the string. When counting by one with the array twenty-nine or
thirty-one wide, I was able to find a string of six primes diagonally. There is a difference
of thirty between the primes in both arrays. Below is a part of the array that shows the
longest string.
4
1
2
3
4
5
6
7
8
9
10
11
12
13
30
31
32
33
34
35
36
37
38
39
40
41
42
59
60
61
62
63
64
65
66
67
68
69
70
71
88
89
90
91
92
93
94
95
96
97
98
99
100
117
118
119
120
121
122
123
124
125
126
127
128
129
146
147
148
149
150
151
152
153
154
155
156
157
158
175
176
177
178
179
180
181
182
183
184
185
186
187
204
205
206
207
208
209
210
211
212
213
214
215
216
Figure 3. Partial 30-column rectangular array string of 6 primes
The equation for this string of primes is y=30x+7, this equation is linear because there is
a difference of 30 between each prime and you add 7 since that is the first prime in this
string. This same pattern also appeared on an array with nine columns, adding three to
each number to get the next.
1
6
11
16
21
26
56
31
36
41
46
51
61
66
71
76
81
86
91
96
101
106
111
116
121
126
131
136
141
146
151
156
161
166
171
176
181
186
191
196
201
206
211
216
221
226
231
236
241
246
251
256
261
266
271
276
281
286
291
296
301
306
311
316
321
326
331
336
341
346
351
356
361
366
371
376
381
386
391
396
401
406
411
416
421
426
431
436
441
446
451
456
461
466
471
476
481
486
491
496
501
506
511
516
521
526
531
536
541
546
551
556
561
566
571
576
581
586
591
596
601
606
611
616
621
626
631
636
641
646
651
656
661
666
671
676
681
686
691
696
701
706
711
716
721
726
731
736
741
746
Another array that also has a string of
six primes was on a six-column array and
adding five to each number. All of the primes
are in the first or third column. The beginning
numbers for these columns are 1 and 11. Since
I was adding five each time all the primes end
with one.
The equation is: y=30x+1, because
there is a difference of thirty between each
prime and you add the initial one from the
beginning of the column. All of the primes are
in the form of mod 6+1 in the first column or
mod 6+5 in the third column.
5
Another array in the form of mod 6+1 or mod 6+5 was a six-column array of
consecutive numbers. On this array all primes fell into the column under one or five with
the exception of 2 and 3. The equation for that array was y=6x-1, (mod 6+5). The
longest string had five primes of 5, 11, 17, 23, 29.
On all of the arrays all of the prime numbers within a string, on the given chart
work out to be either modulo 6+1 or modulo 6+5, but not both within one array. They do
not have the same equations, but the difference between the primes in the string is always
a multiple of six. The vertical string of primes on arrays that I found where this was the
case also include these: 12 column array adding three to each number with an equation
of y=36x-5, longest string had four primes of 31, 67, 103, 139; a 6 column array adding
three to each number with an equation of y=18x+7, longest string had four primes of 43,
61, 79, 97; a 4 column array adding three to each number with an equation of y=12n-5,
longest string had four primes of 7, 19, 31, 43; and a 6 column array of only odd numbers
with an equation of y=12n-7, longest string had five primes of 5, 17, 29, 41, 53. The
horizontal string of primes on arrays where the mod 6+1 or mod 6+1 applies includes: 12
column array of consecutive numbers with an equation of y=12n-7, longest string had
five primes of 5, 17, 29, 41, 53 which is mod 6+5; 10 column array of odd numbers with
an equation of y=18x-1, longest string had four primes of 53, 71, 89. 107 which is mod
6+5; 9 column array of odd numbers with an equation of y=18+7, longest string had four
primes of 43, 61, 79, 97 which is mod 6+1; 9 column array adding three to each previous
number with an equation of y=30x+7, longest string had six primes of 7, 37, 67, 97, 127,
157 which is mod 6+1; 11 column array of odd numbers with an equation of y=24n-11,
longest string had four primes of 79, 103, 127, 151 which is mod 6+1; 6 column array
6
adding four to each previous number with an equation of y=24n-11, longest string had
four primes of 349, 373, 397, 421 which is mod 6+1; 7column array adding five to each
previous number with an equation of y=30x+11, longest string had four primes of 401,
431, 461, 491 which is mod 6+5; 7 column array of consecutive numbers with an
equation of y=6x+1, longest string had four primes of 41, 47, 53, 59 which is mod 6+5;
and 11 column array of consecutive numbers with an equation of y=12x-7, longest string
had five primes of 5, 17, 29, 41, 53 which is mod 6+5.
Spiral Arrays
Another part of my problem was to find the longest string of prime numbers
within a spiral. To spiral the numbers, I started with one in the center then moved out
and around creating a spiral with the numbers. Again, I could change my beginning
number and what I was counting by. Mathematician Stanislaw Ulam first discovered the
spiral array of primes in 1963, while sitting in a meeting. He put the number 1 in the
center and began spiraling out from the center with consecutive numbers. He then circled
the prime numbers and found they tended to line up in a diagonal. Below is an Ulam
spiral that contains a total of 160,000 integers. Prime numbers are the black pixels.
There are 14,683 primes in this Ulam spiral.
7
Figure 5. Ulam Spiral, 1 to 160,000
I began using consecutive numbers with one at the center of the spiral. The
longest string was a string of five primes. They did not go through the center of the
spiral, nor did they have a consistent first difference, so it was not linear nor did it have a
consistent second difference. The second difference was close to being the same, with
only five primes in the string it is hard to be positive that the pattern would continue with
the second differences being eight. If the pattern continued, then the spiral equation
would be in the form of a quadratic. I came up with a quadratic equation that does not fit
the first two numbers in the string, but does fit the last three numbers which is
y=4x^2-4x-1.
8
90
89
88
87
86
85
84
83
82
121
57
56
55
54
53
52
51
50
81
120
58
31
30
29
28
27
26
49
80
119
59
32
13
12
11
10
25
48
79
118
60
33
14
3
2
9
24
47
78
117
61
34
15
4
1
8
23
46
77
116
62
35
16
5
6
7
22
45
76
115
63
36
17
18
19
20
21
44
75
114
64
37
38
39
40
41
42
43
74
113
65
66
67
68
69
70
71
72
73
112
102
103
104
105
106
107
108
109
110
111
Figure 6. Partial consecutive spiral starting with 1.
I then continued with one in the center of the spiral building different spirals by
adding different amounts to one and each previous number on each spiral. I created eight
different spirals with one as the starting point. The longest string I was able to find was
when I added six to each previous number within the spiral; I found a string of thirteen
prime numbers. These primes did not go through the center of the spiral; they were left
of the center. Again, the second difference was not consistent, the second difference of
the first three numbers were different than the remaining primes in the string which had a
second difference of forty-eight. As you can see in the spiral below the bottom two
numbers in the string are not in order. When I dropped the numbers that are out of order,
I could work with the remaining numbers to come up with an equation. The equation for
this string is, y=24x^2+36x+43. The primes in green are all modulus 6+1.
9
3925
3331
2785
2287
1837
1831
1825
1819
1813
1807
1801
1795
1789
1783
3931
3337
2791
2293
1843
1441
1435
1429
1423
1417
1411
1405
1399
1393
3937
3343
2797
2299
1849
1447
1093
1087
1081
1075
1069
1063
1057
1051
3943
3349
2803
2305
1855
1453
1099
793
787
781
775
769
763
757
3949
3355
2809
2311
1861
1459
1105
799
541
535
529
523
517
511
3955
3361
2815
2317
1867
1465
1111
805
547
337
331
325
319
313
3961
3367
2821
2323
1873
1471
1117
811
553
343
181
175
169
163
3967
3373
2827
2329
1879
1477
1123
817
559
349
187
73
67
61
3973
3379
2833
2335
1885
1483
1129
823
565
355
193
79
13
7
3979
3385
2839
2341
1891
1489
1135
829
571
361
199
85
19
1
3985
3391
2845
2347
1897
1495
1141
835
577
367
205
91
25
31
3991
3397
2851
2353
1903
1501
1147
841
583
373
211
97
103
109
3997
3403
2857
2359
1909
1507
1153
847
589
379
217
223
229
235
Figure 7. Partial spiral array starting with 1 and adding 6
The spirals that did not go through the center caused me a lot of trouble. I kept
trying to get my equation to fit all the numbers in the string, and I could not make it
work, because the second differences were not all the same. I asked several people that I
know for help and none of them could come up with an equation that would work. I used
the table that I had created on my spreadsheet, I highlighted what my x and y axis would
be and inserted a chart. Once the chart wizard opened, I selected a scatter plot. When the
chart was created, I right-clicked on one of the data points and selected add trendline, I
used a polynomial trend/regression line and showed the equation. When I finally dropped
the beginning primes and focused on the ones that had a consistent second difference, I
was able to generate an equation that would work for the remaining primes in the string.
I kept creating different spirals with one in the center and never came up with any
with a longer string of primes than thirteen. I finally decided to try different numbers
other than one in the center and see what happened. I started off with placing eleven in
the center and using consecutive numbers, this created a string of only ten primes in a
string. All of these primes had a second difference of two, and the string went through
10
the center of the spiral. I then tried starting with seventeen, I found a string of sixteen
primes, and again all the primes had a second difference of two, and went through the
center of the spiral.
Next I created a spiral starting with thirty-five. This spiral had a
string of twenty primes; the string did not go through the center of the spiral. This was
another spiral that caused me a lot of trouble. The first six primes in the string do not fall
in order because of the spiraling. After the prime 173, the primes all fall in order and
280
281
282
283
284
285
286
287
288
221
222
223
224
225
226
227
228
229
170
171
172
173
174
175
176
177
178
127
128
129
130
131
132
133
134
135
92
93
94
95
96
97
98
99
136
65
66
67
68
69
70
71
100
137
64
47
48
49
50
51
72
101
138
63
46
37
38
39
52
73
102
139
Figure 8. Partial spiral starting with 35.
have a difference of 8. Again, once I dropped the first six prime numbers in this string, I
was able to come up with an equation of y=4x^2-2x+41 that would give me the rest of
the prime numbers in the string.
Another of the spirals I tried started with forty-one in the center, which has a
string of forty primes through the center. This was the longest string that I found. The
second difference of this string is two and the equation is y=x^2-x+41.
11
Mathematical Ideas
It appears that when the string of primes goes through the center, the second
difference remains constant throughout, when it does not go through the center the
second differences are not constant in the beginning. The four spirals where I started
with numbers other than one in the center all have the same pattern when looking at
modulus 6 which is: mod 6+5, mod 6+1, mod 6+5, mod 6+5, mod 6+1… After finding
a modulus pattern in the rectangular arrays first, I then looked for modulus patterns in the
spirals, which I found. Although there are patterns within modulo and primes, there are
no patterns within primes themselves. There is no one equation that will generate prime
numbers. Within all of the different arrays I created, there are an equal number of
equations, with each equation specific to a certain array. There are similar forms of
equations, but no one specific equation will work for all.
12
Bibliography
Charpentier, Michel. (2001) Prime numbers in PostScript. Retrieved from
http://www.cs.unh.edu/~charpov/Programming/PostScript-primes/#first
Marteinson, Peter. “Observations on the Regularity of Prime Distribution”. AS/SA No
14, Article 7. Retrieved from http://www.chass.utoronto.ca/french/as-sa/ASSA14/article7en.html
Peterson, Ivars. Prime Spirals. Science News Magazine of the Society for Science & the
Public. Retrieved from
http://www.sciencenews.org/view/generic/id/2696/title/Prime_Spirals
Rudd, H. (2010) UlamSpiral.com. A visual analysis of prime number distribution.
Retrieved from http://ulamspiral.com/generatePage.asp?ID=1
Weisstein, Eric. (1999-2010) Prime Sprial. WolframMathWorld Research. Retrieved
from http://www.maa.org/mathland/mathtrek_05_06_02.html
Wells, David. (2005) Prime Numbers The Most Mysterious Figures in Math. Hoboken,
NewJersey: John Wiley & Sons, Inc.
Uittenbogaard, Arie. (2000-2007) 8.2 There’s Something About the Number Sequence –
Ulam’s Rose or Prime Number Spiral. Retrieved from http://www.abarimpublications.com/artctulam.html
Figure 2: http://www.chass.utoronto.ca/french/as-sa/ASSA-14/article7en.html
Figure 5: http://mathworld.wolfram.com/PrimeSpiral.html
13
Appendix
primes.xls
14